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shortening the results of previous experiments by a third, and
lengthening them by an eighth,
and here was the be-
ginning of sorrows.
The attempt (beyond these three
steps) at the division of the "Canon"-in other words, at the
division of a string into the lengths which produce the sounds
that make music in a single key-was a failure."

The experiments of modern philosophers have been rewarded with the discovery that a musical string divided in the proportions given underneath will produce the notes of the scale as there described. Let it be noticed that the figure 1 stands for the whole length of the string, whether a foot, a yard, or any other measure, and whatever sound (in pitch) it gives-that Bound being taken for the key note-Dон. It may also be mentioned that the same numbers denote the comparative lengths of organ pipes capable of sounding the corresponding

notes.

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Perhaps these proportions will be better understood by the annexed diagram. A single string thus stretched and used for these experiments is called a monochord. If the student is of a mechanical turn, let him make one and verify the measurements here given. Let him suspend a board of four or five feet in length against a wall. To the upper part of this board fasten the end of a pianoforte-wire or other musical string which is of the same thickness throughout. Let the wire pass down the face of the board, over a firm wooden bridge, an inch or so high, and close to the top, and over a movable bridge at the bottom; and let it be kept stretched by a heavy weight. Set your movable bridge (which the weight will keep in its place) at the bottom, marking the spot, and take the sound of the whole string, by the help of a fiddle bow, for your Doн, or key-note. Then (having properly measured and marked the board) move the bridge to the other divisions, sounding each note as before. It may be well to mention that Colonel Thompson maintains, and with good show of reason, what he calls the " duplicity" of RAY and TE. They are sometimes sounded by good singers and violinDOH players a very small degree lower than their usual position given above. These experiments will fix in your mind a clear notion of the scale.

ΤΕ

LAH

FAR

RAY

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If our arithmetical friend will now work a few sums in proportion, he will be able to show the value of the intervals between the several notes of this scale. Thus the vibrations of Dон differ from those of RAY, in being three less, and (three being one-ninth of twenty-seven) DOH has therefore only eightninths of RAY's vibrations. The same proportion will be found between FAH SOн, and LAH TE. These intervals are called the great tones." The proportion of RAY ME, and of SOH LAH is nine-tenths. These are the "small tones." The proportion of ME FAH, and of TE DOн, is fifteen-sixteenths. These are called semitones, or, more properly, Tonules. If you calculate from the length of the string given above you will find still the same proportions existing.

66

Let our arithmetical friend reduce these "ratios," or proportions, of the three intervals in the scale to fractions having a common denominator. They will then stand thus:-

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Now this evidently means that the lower note of the "great tone" has 1,280 vibrations, while the higher note has 1,440, and (as the lengths of string are in inverse proportion to the vibra tions) that it takes 1,440 degrees of the string, while the higher takes only 1,280 such degrees. Therefore the proportional difference between them, whichsoever way you look at it, is one hundred and sixty degrees. In the same way you will find that the difference between the two notes of this "small tone" is one hundred and forty-four degrees, and that the interval of the "tonule" is ninety degrees. The degrees in each case are of similar value, all measured on the same scale (common denominator) of 1,440 degrees. We may therefore treat them as belonging to one scale, and adding three "great tones," two "small tones," and two "tonules" together, we shall obtain a perfectly measured scale of 948 degrees. As all these numbers, however, will divide by 2, retaining, of course, the same proportion to one another, it is better to regard the scale as comof 474 degrees, containing three " great tones" of 80 degrees, two "small tones" of 72 degrees, and two "tonules" of 45 degrees, and this is the smallest perfect measurement of the scale in plain figures. But if the pupil will go one step further, and divide each of these intervals by nine, he will see how we obtain the proximate scale of fifty-three degrees. The tonule will be exactly 5 degrees, the small tone exactly 8 degrees, and the great tone only one-ninth of a degree less than 9 degrees. Adding these together, as before, you will have the "Index scale," as Colonel Thompson calls it, "of fifty-three," and you will see that it is three-ninths or one-third of a degree too large. We strongly advise the pupil to construct a chord," and try for himself whether this is not in truth an accurate description of that scale of related notes which God has made most suitable to human ears and souls. All the books of science are agreed that it is; and experience bears the same testimony. It is the more important that you should understand these points, because the true scale is dreadfully abused by the common keyed-instruments. Many of these are tuned by what is called "equal temperament;" that is, the scale is divided into twelve equal semi-tones, and it follows that the tones are all 79 degrees (of the perfect scale of 474), while they ought to be sometimes 80 and sometimes 72 degrees! and the tonules (semitones) are both 39 instead of 45!! They might as well cut down the fingers of a statue to "equal temperament!" Human ingenuity will surely deliver us soon from this me strous distortion. You will understand now why it is so pleasanter to sing "without the pians."

It will be well for you to understand the connection between these musical notes and the vibrations of the sonorous body which produces them-posed Son whether that body be the string of a violin, the air in an organ pipe, a small plate of glass or metal, or the "chorda vocales"-the vocal chords of that wonderful little box instrument, called the " larynx," which you can feel in your own throat. Sounds ME produced by irregular vibrations are not musical. They form the "roar, rattle, hiss, buzz, crash," or some other noise. But sounds produced by equal and regular vibrations are musical. "That musical notes are produced by a rapid succession of aerial impulses at equal intervals, is very clearly illus. trated by an instrument called the syren, the inDON vention of Cagniard de la Tour. A blast of air is forced through a narrow aperture in a pipe; and a flat circular disk, perforated near its circumference STRING OF with a number of small holes equidistant, and in a circle concentric with the disk, is so applied to the pipe, that the blast is interrupted by it, excepting when one of the holes in the disk is opposite to that of the pipe; and when the former is made to revolve rapidly, the resulting aerial impulses cause a series of isochronous vibrations that produce a musical note, and the corresponding number of its vibrations can very easily be computed, from knowing the number of holes and of revolutions of the plate. The results obtained by this instrument agree exactly with those found by other methods." The more rapid the vibrations of the sonorous body, the more "acute" (shriller, or higher) the note produced.

IIVISIONS

OF THE

A MONO

CHORD.

mono

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er exercises, are to be sung twice as

notes; and the open notes without a

1. this exercise, are to be twice as long

is CORE FEL & STs relative length does not, however, Teme te. An open note in one tune may Jaark note in another, and a black note in 21ere to open note in another. Let it, howAPA be much better for the learner not And it present to the old "notation" (way of TID POWs thas placed between brackets. He By: pamed with the notation of music, when he le attention to music itself. Sing es and never leave an exercise until memory, pointing on the

T= sety to

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load to that as you sing them, at every mistake or uncertainty striking the oled" and beginning again with great patience. When you Than sol the chant from the modulator by memory, then learn the to us, instead of the syllables, the words "One, two, three, Sur; one, two, three, four, five, six," still pointing to the right Termotes on the modulator. It may be well for you now to learn we Ms the second line of notes (to be sung by another voice along with www can the upper line) as you learnt the first, and not, for the present, we attempt the words. But if you wish to use the words, then first by the mark har to sing the words "Trust in the Lord with " on the single to the motor mote Me. To do this with distinct utterance, you should divide

the note (in your mind) into two equal beats or "pulses" of time (you can mark them by beating on the table with your hand); and then the words "Trust in the" will go to the first beat, and "Lord with" to the second. The large dot above the line shows this division. The words "all thine heart" fall easily to their right notes. To the note Soн you will sing the words" and lean not unto thine." "And" is scarcely heard. Dividing Soн, like the other "reciting note" ME, into two beats (and reciting notes of a chant like this may be divided into as many beats as you please), you will have the words "lean not unto" to the first beat, and "thine" alone filling the second. The word "own" you perceive is "slurred" on to two

un

notes. Take care not to sing the syllables "standing" quickly and sharply. Let them take as much time as the syllables "under" in the same word. The second verse of words, printed underneath, is divided on the same plan as the first. The double bar, you will observe, separates the words of the " reciting note" from those of the "cadence" (as it is called) of the chant. On the upper "staff" you miss the square note for Dон, because the tune begins on ME, and Don does not occur in the "air." The place of Doн, however, is in the first space, reckoning from the bottom of the staff. On the lower staff it was necessary to make an additional line to carry Soн. This is called a "ledger line."

COPY-SLIP, NO. 15.-COMBINATION OF THE LETTERS u, n.

im

COPY-SLIP, NO. 16.—COMBINATION OF THE LETTERS i, m.

nut

COPY-SLIP, NO. 17.-COMBINATION OF THE LETTERS n, u, t.

nit

COPY-SLIP, NO. 18.--COMBINATION OF THE LETTERS n, i, 1.

tun

COPY-SLIP, NO. 19.-COMBINATION OF THE LETTERS t, u, n.

LESSONS IN PENMANSHIP.-VI.

In our last lesson we showed our learners how to make the letters m and n by combinations of the top-turn and the topand-bottom-turn. In the present lesson we set before them five Copy-slips for practice, comprising various combinations of the letters m and n with the letters i, u, t, and 1, which they learned to form from the simple bottom-turn. Doubtless many of those who have been endeavouring to teach themselves the art of Penmanship by the aid of our lessons, are now beginning to see that, after all, it is not such a difficult matter to learn to write, since by learning to form only three strokes of a very simple kind, they have found that they have acquired the power of writing six out of the twenty-six letters that form the alphabet. They will soon see that this is by no means the utmost limit of their progress, when in the course of future lessons they discover how many letters there are into whose composition these three elementary strokes also enter.

In preparing paper for copying all our clementary copy-slips in large text, the learner must not omit to insert the diagonal lines as they appear in Copy-slips Nos. 1 to 6 inclusive, as long as he finds them absolutely necessary for the regulation of the slope of his letters, and to enable him to preserve a proper distance between them. To save trouble in ruling sheet after sheet of paper with horizontal and diagonal lines at the proper distances from each other, and to save expense as well, the learner might rule with a sharp-pointed steel instrument, such as a bradawl, one side of a cheap slate with sets of lines similar to those in our copy-slips, taking care, however, to leave a space of one inch and a quarter between each set, to enable him to add the extra horizontal lines that will be required when he begins to make looped letters, and letters such as p and q, that extend below the lower of the two horizontal lines that contain what we have called the body of the letter. A slate thus prepared will be found useful for acquiring facility in forming the various letters, but the learner must by no means omit to write on paper as well.

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(See

Reif, ripe.
Silbern, silver.
Tin'tenfaß,n.inkstand.
Tisch, m. table.
Tischler, m. joiner.
Better, m. cousin.
Weter- noch, neither

--nor.

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5.

1. Haben Sie mein reifes Obft? 2. Nein, ich habe Zbr silbernes CSA meffer, und Ihr alter Freund, der Lebrer, hat das reife Ch 3. Haben sie meinen silbernen Bleistift? 4. Nein, der gute Sebrer bat ihn. Hat der alte Koch meinen bölzernen. Tisch? 6. Nein, der Tischler bat ihn, aber ter Koch bat einen marmernen Tüch. 7. Hat er auch ein bölzernes ทั้งนี้ ? 8. Ja, und dieser fleisige Schüler hat ein schines, bleiernes Tinten fas 9. Hat er auch einen ülbernen Becher? 10. Ja, und er bat auch einen fupernen Kefel und einen eisernen Norier 11. Haben Sie das neue Messer meines jungen Freundes? 12. Neiz at babe ein neues Messer von bem quten Kaumanne 13. Sit dieet knjige Schüler das gute Buch des alten Freundes, eder den ülbernen. Burdet innes zuten. Wetters? 14. Or bat weder ein extes Bus. 250 enen Übernen Stahit—er bat nur einen bölzernen Bleck.t 13. Better to Section IX. 2) Kennel bed Goda? 16. Der arme Rama de ret ernen erjernen. Keñel

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roth-er, to or for red; roth-e, red.

When preceded by either of the articles, or by any one of the adjective pronouns (see lists Sect. IX. and X.), the adjective terminates in the nominative and accusative as in the old declension, but in the genitive and dative in the letters cn. Thus :

N. tie gut-e, the good;

G. der gut-en, of the good;
De ter gut-en, to or for the good;
A. rie gut-e, the good;

meine alt-e, my old; meiner alt-en, of my old; meiner alt-en, to my old; meine alt-e, my old.

I. The personal pronoun Sie (you) is always written with a capital initial, while fte (she or her) is only thus written at the beginning of a sentence. Hence in writing, no ambiguity can arise. Ex., 3ch sebe Sic, I see you; ich sehe sie, I see her. When Sic is used in the nominative, the form of the verb determines the person. Ex., Sie sehen ihn, you see him; Sie sicht ihn, she sees him. Whether, however, ic (when in the accusative) stands for you or her, can only be determined by the context. The orthography of the possessive pronouns 3hr (your) and ihr (her) Thus, 3hr Buch tit gros, may signify, your book is large, or her is also identical, and, in speaking, is liable to equal ambiguity. book is large; and, Ich habe ihr Buch, may mean, I have their accusative, and of ibr in all the cases, must of course, when book, or I have her book. The significations of fie in the spoken, be determined by the connection. (See Declension, Sect. XVIII.)

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RESUME OF EXAMPLES.

'vernglas, n. operaglass.

Scheere, f. scissors. Schwester, f. sister. Seite, f. silk. Tante, f. aunt. llbr, f. watch, clock. Ubr'tasche, f. watchpocket.

Der Bruter hat das Buch ter The brother has the book of

Schweiter

Der Vater giebt ter Tochter en Much

Der Hut meiner Mutter ist schön.

et vie Uhr Ihrer Fräulein

Gouine?

Sie ist in der Hand ihrer Mutter.

the sister.

The father gives the daughter a book.

The hat of my mother is beautiful.

Where is your cousin's watch?

It is in her mother's hand.
EXERCISE 14.

1. Iû rie Junge Schwefter dreier jungen Dame in Deutschland? 2. Nein, fie it in Frankreich aber ihr Bruter ià in America 3. We ist meine neue, göldene Feder? 4. Ihre junge Freuntin Fräulein bat sie (Sect. XVIII. 3.) 3. Hat Jöre Mutter rie ichine Seite Ihrer Tante? 6. Ja, und auch die schöne, feine Seinwand 7. We in Zbre goldene Brille ? 8. 3 babe keine geltene Brille 9. Haben Sie eine fiberne, eter eine goldene Ubr? 10. Z& babe eine sliderne libr. 11. Ju fie eine gute Übr? 12. Ja aber e vikt mcht sehr ichim. 13. Be in Ihre Ute? 14. Sie ist in meiner Uhrtaiche 15. Hat Ihre Schweitet eine geldene Ubr? 16. Ja, und he bat auch eine schöne geltene Seite. 17. Be it meine neue berre 18. 3d babe sie aber fie in macht sehr scharf. 19. Be it Ihre Swener? 2). Sie våt bei der (Sect. XVII. 3.) Nutter in der bactdek. 21. Be in meine aläjerne Samre? 22. Ich habe ne. 23. Wer hat mein neues Coerngias * 24. 3d babe es und Ihre neue

Tyminine nouns are in the singular indeclinable; as, nom. tue Brule Svire (the milk) ; gen der Seite; dat. der Seite, ace, die Seite

The adjective in the romanèse slagular has two forms. When it stands alone, or nnadfected by a preceding word (§ 291, the nominative and accusative end in e, the genitive and dative in er It is then eard to be of

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7

EXERCISE 15.

1. The mother of this lady is in France. 2. Has the beautiful daughter of the good ter geten azt a golden (geltent") watch? 3. My diligent brother has neither a golden watch nor a good entes opera-class. 4. Mr good sister has no feine fine linen, but aber she has a new near glass lamp 5. My cousin with at the (ter) golden spectacles is with [be] my [meinen] beautiful brother in the library,

LESSONS IN ARITHMETIC.-VI.

ABRIDGED METHODS OF MULTIPLICATION AND DIVISION. 1. THE methods of multiplication and division explained in the previous lessons are those ordinarily employed; and the learner must make himself perfectly familiar with them before proceeding further.

These processes, however, in particular cases, can often be materially facilitated by various artifices. Some of these shorter methods we subjoin, not only because they are useful in themselves, but because they are valuable as exercises, in explaining the fundamental principles of arithmetic.

2. Any number which is formed by multiplying two or more numbers or factors together is called a composite number.

352

It

has already been explained in a former lesson that the same numbers multiplied together will give the same product, in whatever order the multiplication is effected. Hence, to multiply any number by one which is compositei.e., which is composed of several factors-we have only to multiply the number first by one factor, the result by another factor, and so on. Thus, to multiply 352 by 28, since 287 x 4, we can perform the operation as indicated in the margin.

7

2104 4

9855 = 28 × 352

2, 35, 54, 56, 63.

EXERCISE 10.

3, 45, 72, 64, 81, 96.

(1.) Resolve the following sets of numbers into their factors: 1, 9, 10, 14, 22. 4, 8, 16, 18, 20, 24. 5, 27, 32, 36, 40, 48. 6, 12, 28, 51, 72, 84. (2.) What will 24 horses cost at 74 crowns apiece. (3.) What will 45 hogsheads of tobacco cost, at 128 crowns a hogshead?

(4.) What will 54 acres of land cost, at 150 crowns per acre? (5.) At 118 shillings per week, how much will it cost a family to board 49 weeks?

(6.) If a man travel at the rate of 72 miles a day, how far will he travel in 64 days?

(7.) At 163 crowns per ton, how much will 72 tons of lead cost?

(8.) What will 81 pieces of broadcloth cost, at 245 shillings apiece?

(9.) What will 84 carriages cost, at 384 crowns apiece? (10.) What will a railway 132 miles in length cost, at the rate of £1,960 a mile ?

(11.) If I can walk a mile in 16 minutes, how long will it take me to walk 374 miles ?

3. Similarly, it will be seen that to divide by any composite number, we have only to divide by one factor, then divide the quotient by another factor, and so on.

9856

28

1408

352 Answer.

Thus, to divide 9856 by 28, arrange the process as indicated in the margin.

In this case there is no remainder. But suppose it be required to divide 9873 by 28.

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(2.) How many acres of land, at 35 crowns an acre, can you buy for 4650 crowns.

(3.) A man divided 837 crowns equally among 27 persons, who belonged to three families, each family containing nine persons: how many crowns did each person receive ?

(4.) A man bought a quantity of clover seed amounting to 507 pints, which he wished to divide into parcels containing 64 pints each: how many parcels can he make?

5. Multiplying and dividing by powers of 10, and by numbers ending in any number of ciphers.

The products of two tens, three tens, four tens, etc., are called respectively the second, third, fourth, etc., powers of 10. They are 100, 1000, 10000, etc. Thus, the second power is 1 followed by two ciphers, the third 1 followed by three ciphers, and so on; the number of the ciphers in each case being the same as that of the power.

It has been already explained that to multiply by 10, or any power of 10, we have only to annex to the multiplicand the number of ciphers corresponding to the power. Thus, 345 multiplied by 1000 is 345000.

If any number of the right-hand figures in the multiplier be ciphers--as, for instance, in 75000-then, as we have already Seen in Lesson IV., Art. 5, we need only multiply the multiplicand by 75, and annex to the product the same number of ciphers, in this case three.

EXERCISE 12

(1.) Work the following examples in multiplication:

1. 153486 × 10000.
2. 3120467 x 100000.

3, 52690078 x 1000000.

4. 689063157 × 10000000.

5. 4946030506 × 100000000.

6. 87831206507 × 1000000000. 7. 67856005109 x 10000000000. 8. 14376 × 25000.

9. 350634 x 410000. 10. 4630425 × 6200000.

11. 2370000 × 52. 12. 48120000 × 48. 13. 356300000 × 74. 14. 1623000000 × 89. 15. 540000 × 700. 16. 1563800 × 20000. 17. 31230000 × 120000. 18. 5310200 × 3400000. 19. 82065000 x 8100000. 20. 210909000 × 5100000.

(2.) What will 10 boxes of lemons cost, at 63 shillings per box?

(3.) How many bushels of corn will 465 acres of land produce, at 100 bushels per acre?

(4.) Allowing 365 days for a year, how many days are there in 1000 years?

(5.) How much will 50 hogs weigh, at 375 pounds apiece? (6.) If 1 barrel of flour weighs 192 pounds, how much will 500 barrels weigh?

LESSONS IN GEOMETRY.-III.
INSTRUMENTS USED IN PRACTICAL GEOMETRY.

IN the operations of practical geometry, a case of mathematical instruments must be considered as an essential requisite. These instruments vary in number and quality, according to their price. Some are made of wood, bone, and ivory-as rulers and scales; cthers are made of brass and steel, German

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